Wayne Routledge's early goal looked set to give Paul Clement's side all three points but the pressure finally told with two minutes remaining when Dele Alli tucked home Christian Eriksen's deflected cross.
Heung-min Son and Eriksen both struck deep into seven minutes of injury time to seal a dramatic comeback for Mauricio Pochettino's men.
The north Londoners have coped admirably without several key players in recent weeks and their squad - tested to the limit in South Wales - once again proved its worth.
Spurs needed a matter of seconds to fashion their first opportunity as Son exchanged passes with Kyle Walker before seeing his effort well blocked.
Swansea went down to 10 men temporarily when Jack Cork was forced off with a head wound and Moussa Sissoko should have made the hosts pay after being teed up by Son, but the Frenchman's composure inside the penalty area failed him.
The hosts got a foothold in the game courtesy of a series of corners but it was a long punt upfield by Lukasz Fabianski which resulted in an early breakthrough.
Jordan Ayew raced onto his goalkeeper's clearance before bamboozling Ben Davies at the byline and squaring for Routledge, who slotted home from close range.
Alli almost hit back immediately with an audacious acrobatic effort while Eriksen's long-range strike kept Fabianski on his toes.
But the north Londoners' usually stellar defensive organisation was scarcely seen during the first half, with Ayew giving Davies and Co. plenty of problems as he strutted around the Liberty Stadium pitch.
Son was full of running up top and it was perhaps surprising Pochettino did not shuffle his pack at the interval, having seen his side struggle to make significant inroads in the final third of the pitch.
The Argentine coach did switch his system, with Spurs reverting to a back three consisting of Eric Dier, Jan Vertonghen and Toby Alderweireld.
The Swans, meanwhile, were content to sit back and get men behind the ball out of possession.
As such Spurs dominated but had a paucity of gaps to exploit and the first real signs of frustration set in on the half hour mark when Dier abandoned a piece of patient build-up play by lashing a long-range effort well wide.
Vincent Janssen was called upon soon after, giving Spurs some much needed presence up front.
Swansea's resilience grew as the final whistle approached, with Fabianski denying Alli despite carrying an injury.
But the 20-year-old made no mistake when Eriksen's shot fell into his path while Son showed superb composure to slot home the north Londoners' second strike of the evening.
And, with Swansea out on their feet - seemingly unable to believe how they had had their pocket picked, Eriksen swept home to complete an incredible turnaround.
Pochettino's men simply do not know when they are beaten. The title race is alive - just.